More complete discussion of the time-dependence of the non-static line element for the universe

In a previous article,(1) I have shown that a continuous transformation of matter into radiation, occurring throughout the universe, as postulated by the astrophysicists, would necessitate a nonstatic line element for the universe, and have shown that the non-static character thus introduced might provide an explanation of the red shift in the light from the extra-galactic nebulae. In the present article, I wish to discuss the form of dependence of the line element on the time more completely than was possible on the previous occasion. This is a matter of considerable importance, since changes in the approximations which must be introduced to obtain a usable result affect to quite a different extent the expressions for the relation between red shift and distance and for the rate of annihilation of matter. Indeed, the possibility arises of slight changes from the treatment previously given which would leave the theoretical relation between red shift and distance still approximately linear, as observationally found, and yet produce a very considerable change in the calculated rate for the annihilation of matter

Topological properties of Hamiltonian circle actions

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of \pi_1(\Ham(M,\om)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of \pi_1(\Ham(M,\om)). Further, if the isotropy has order at most two and the circle contracts in \Ham(M,\om) then the homology of M is invariant under an involution. For example, the image of the normalized moment map is a symmetric interval [-a,a]. If the action is semifree (i.e. the isotropy weights are 0 or +/- 1) then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of $M$ to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov--Witten invariants on symplectic manifolds with S^1-actions.Comment: significantly revised, some new results added, others moved to SG/0503467, 56 pages, no figure

New Techniques for obtaining Schubert-type formulas for Hamiltonian manifolds

In [GT], Goldin and the second author extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99] and [Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $\alpha_p$ in the equivariant cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these classes are the equivariant Schubert classes. It is a long standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class $\alpha_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.Comment: v2; Significant revision. 52 pages, 1 figure. To appear in Journal of Symplectic Geometr